Understanding Transvections:T_W:V->V Tv=v+w

[Bacle]

Understanding Transvections:T_W:V-->V Tv=v+w

Esteemed Algebraists:

Please help me understand better the definition of a transvection.

Let V be a finite-dimensional vector space, and let W

be a codimension-1 subspace of V . A transvection

is defined to be an invertible linear map T:V-->V

such that:

i) T|_W =1_W , i.e., the restriction of T to W

is the identity on W.

ii)For any v in V, T(v)=v+w ; w in W.

Condition i) is clear, but does condition ii) just say that

vectors in V-W are mapped to V-W?

Also: given a choice of basis for V, is the

matrix representation for V always that of

a shear matrix, i.e., a matrix with all diagonal

entries equal to 1, and all off-diagonal entries

except for exactly one equal to zero, i.e., a

matrix describing adding a multiple by k of one

row to another row?

I know this is the representation in vector spaces over R; is

it true for V.Spaces over any field F? ( I know all V.Spaces of same

dimension are isomorphic, but I don't know if that guarantees the result).


I was thinking of a simple example of a linear map from

R³ to R³

preserving points of types (x,0,0) and (0,y,0). Then ii) above would say that, using the

standard basis {e__i; i=1,2,3}.

i) T(1,0,0)=(1,0,0)

ii) T(0,1,0)=(0,1,0)

iii) T(0,0,1)= (0,0,1)+(a,b,0) ; a,b in F


Is the intended meaning that for z in V-W, T(z) in V-W? Also, the representation of

this transvection does not seem to match that of a shear transformation, since it includes

the case of two non-zero entries a,b.

Any Ideas?

Thanks in Advance.

Thanks.

 

[micromass]

Esteemed Algebraists:

Please help me understand better the definition of a transvection.

Let V be a finite-dimensional vector space, and let W

be a codimension-1 subspace of V . A transvection

is defined to be an invertible linear map T:V-->V

such that:

i) T|_W =1_W , i.e., the restriction of T to W

is the identity on W.

ii)For any v in V, T(v)=v+w ; w in W.

Where did you find this definition. I don't say it's incorrect, but it's a bit weird. The definition on wiki is a little better:

To be more precise, if V is the direct sum of W and W′, and we write vectors as

v = w + w′

correspondingly, the typical shear fixing W is L where

L(v) = (w + Mw′) + w ′

where M is a linear mapping from W′ into W.

I like it better because it's little less ambiguous.

Condition i) is clear, but does condition ii) just say that

vectors in V-W are mapped to V-W?

No. Of course, the definition will still imply that V-W is mapped to V-W, but that's not quite enough. Take V=R^2 and W=R. Then T(1,0)=(1,0) and T(0,1)=(0,2) will satisfy that V-W is mapped to V-W, but it's not a transvection since there is no w in W such that T(0,1)=(0,2)+w.

Geometrically, a transvection maps W onto itself and translates points outside of W parallel to W.

Also: given a choice of basis for V, is the

matrix representation for V always that of

a shear matrix, i.e., a matrix with all diagonal

entries equal to 1, and all off-diagonal entries

except for exactly one equal to zero, i.e., a

matrix describing adding a multiple by k of one

row to another row?

I don't think this is true. If we follow wiki's definition, then a transvection always has the form

$$\left(\begin{array}{cc} I & M\\ 0 & I\end{array}\right)$$

In my (limited) understanding of the topic, a shear matrix will represent a transvection, but not vice versa.

You could help me enormously in providing the reference you're using...

 

[Bacle]

Hi, Micromass:

Sorry for the delay. I am going from the book "Classical Groups

and Geometric Algebra", by Larry C. Groves; a GTM book; mostly

pages 7 and 22. He describes transvections on a fin.-dim v.space

V; with invariant codimension-1 subspace W, as maps T:V-->V ,

with T|_W =1_W (i.e., map T:V-->V

restricts to the identity on W ), and , for any v in V, T(v)=v+w for

some w in W.

Grove goes on to show that transvections generate

both SL(V):={M in GL(V), Det(M)=1}, as well as the

symplectic group of V, given a symplectic form.

 

[Bacle]

Where did you find this definition. I don't say it's incorrect, but it's a bit weird. The definition on wiki is a little better:



I like it better because it's little less ambiguous.



No. Of course, the definition will still imply that V-W is mapped to V-W, but that's not quite enough. Take V=R^2 and W=R. Then T(1,0)=(1,0) and T(0,1)=(0,2) will satisfy that V-W is mapped to V-W, but it's not a transvection since there is no w in W such that T(0,1)=(0,2)+w.

Yes, I understand that; but , since a transvection does map elements in V-W to elements in V-W, what additional condition do we need to characterize transvections? Maybe better to leave good-enough alone and accept T:v=v+w. So what else does the def. say that T:v-w is sent to v-w?

Geometrically, a transvection maps W onto itself and translates points outside of W parallel to W.

Well, actually, you would need a notion of orthogonality defined on your space; there are abstract vector spaces without an inner-product (thing homology over Z/2), without a standard ( if at all) notion of orthogonality.
.

I don't think this is true. If we follow wiki's definition, then a transvection always has the form

$$\left(\begin{array}{cc} I & M\\ 0 & I\end{array}\right)$$

In my (limited) understanding of the topic, a shear matrix will represent a transvection, but not vice versa.

You could help me enormously in providing the reference you're using...

Sorry to bring this up so late, I was just reviewing my posts.

 

[blue_raver22]

I can provide a response to your question about transvections. A transvection is a type of linear map that is defined on a finite-dimensional vector space V, with a codimension-1 subspace W. This map, denoted as T_W:V->V, has two conditions that must be met in order to be considered a transvection. The first condition, i), states that the restriction of T to W is the identity, meaning that the map leaves all vectors in W unchanged. The second condition, ii), states that for any vector v in V, T(v) is equal to v plus a vector w in W. This means that the map will move vectors in V-W to other vectors in V-W, while leaving vectors in W unchanged.

To better understand this definition, let's consider an example in R^3. If we have a linear map T that preserves the points (x,0,0) and (0,y,0), then condition ii) tells us that for any vector (0,0,z) in V-W, T(z) will also be in V-W. This means that the map T is a transvection.

Regarding the representation of transvections, it is true that in vector spaces over any field F, the matrix representation of a transvection will be a shear matrix, with all diagonal entries equal to 1 and only one off-diagonal entry equal to 0. This is because all vector spaces of the same dimension are isomorphic, meaning that they have the same underlying structure. Therefore, the representation of a transvection will be the same regardless of the field F.

I hope this helps clarify your understanding of transvections. If you have any further questions, please don't hesitate to ask. Keep exploring and learning!